Transcript The necessity of identity

The necessity of identity
Kripke versus Gibbard
1. Kripke on the necessity of identity
• Kripke gives the short proof of the necessity of identity at the
beginning of his seminal article (1971/1979, 136)
• Kripke gives credit to Wiggins (1965)
• However, the conclusion was derived earlier by Ruth Barcan Marcus in her “Identity of individuals in a strict functional calculus of second
order” (series of three papers)
• In the present form was first given by Quine (Burgess 2014)
The argument
1. ∀x∀y (x = y → (Fx → Fy))
2. ∀x □x = x
- indiscernibility of identicals
- universal necessary self-identity
3. ∀x∀y (x = y → (□x = x →□x = y)) - instance of (1)
4. ∀x∀y (x = y →□x = y)
- necessity of identity
Derivation
5. ∀x □x = x
Premiss
6. ∀x∀y (x = y → (□x = x →□x = y))
Premiss (instance of Lebniz’s Law)
7. ∀x∀y (□x = x → (x=y →□x = y))
(6) Equi.
8. ∀x∀y (x = y →□x = y)
(5), (7) MPP
Presuppositions
2. ∀x □x = x
- universal necessary self-identity
• The most important premiss, postulated by Kripke
• Weak modal reading - metaphysical modality:
• “Let us interpret necessity here weakly. We can count statements as
necessary if whenever the objects mentioned therein exist, the statement is
true.” (Kripke 1971, 480)
• Kripke: acceptance of the weak reading of (2) commits one to the
conclusion (4)
Conclusion – formula (4)
4. ∀x∀y (x = y →□x = y)
- necessity of identity
• Kripke says that formula (4):
• “does not say anything about statements at all. It says for every object x and
object y, if x and y are the same object, then it is necessary that x and y are
the same object.” (Kripke 1979, 480)
Modality de re versus de dicto
• If the modality were construed de dicto the argument would be
invalid
• De dicto modality – Quine’s example (Quine 1961)
9. ‘number 8 is greater than 7’ - necessarily true
10. ‘the number of planet is greater than 7’ – contingently true
• 9. and 10. are extensionally equivalent – pick out the same number
• truth of the statement can depend on how the referent is picked out
• If picked out by the proper name, such as ‘number 8’ then the statement can
express a necessary truth
• If picked out by the description ‘the number of planets’ then it does not
express a necessary truth.
Why Kripke does not prove premiss 2.?
• Informally it seems straightforward to derive (2)
11. ∀x x = x
11a. a=a.
11b. □a=a
2. ∀x □x = x Universal generalization from 11b. since a is arbitrary
Problem: the derivation presupposes Barcan
formulae
• Barcan schema:
• ∀x □Fx → □∀x Fx
(∃x ◊x=x → ◊ ∃x ◊ x=x)
11. ∀x x = x
12. □∀x x = x
13. □ ∀x x = x → ∀x □x = x
2. ∀x □x = x
Converse Barcan Schema:
□∀x Fx → ∀x □Fx
(◊ ∃x x=x → ∃x ◊ x=x)
Premiss
11. Rule of necessitation
CBS
12., 13. → elim. (Burgess 2014, 1573)
Problem: the necessity of existence
• Barcan Schemas – allows the proof according to which
• everything that could exist does exist, and everything exists necessarily
(Hayaki 2006)
• In particular Timothy Williamson (2002) has argued that he exists
necessarily
• BS ∃x ◊x=x → ◊ ∃x x=x is equivalent to □∀x Fx → ∀x □Fx
• CBS ◊ ∃x x=x → ∃x ◊ x=x is equivalent to ∀x □Fx → □∀x Fx
2. Defense of the necessity of identity
• Kripke examines three different kinds of identity claims and explains
in what way they could be contingent and why if they are contingent
do not count against the conclusion of the argument that all things
are necessarily identical to themselves
Three types of identity statements
A) Identity statements that relate individuating descriptions: such as
“the inventor of bifocals is identical with the first Postmaster general of
United States”.
B) Identity statements that relate proper names: such as “Cicero is
Tully”.
C) Identity statements that pertain to theoretical reductions in science:
such as “pain is identical to firing of C-fibers” or “Heat is mean kinetic
energy”, etc. (Burnyeat 1979)
• A) type of identity statements can be contingent – this is consistent
with 4)
• Example – Russell’ theory of descriptions (scope of descriptions)
• „Just one thing x was the first Postmaster General of the United
States and just one thing y was the inventor of bifocals and it is
necessary that x = y” (Burnyeat 1979, 472)
• Formally:
14. [ὶx Px] & [ὶy Iy] & □x=y
• B) and C) denote necessary identity claims
• Proponents of contingent identity make the mistake of confusing
metaphysical and epistemological notions
• Hesperus=Phosphorus – discovered a posteriori (epistemology)
- necessary identical (metaphysics)
Proper names – rigid designators
• Proper names refer rigidly:
• “a term that designates the same object in all possible worlds.”
(Kripke, 1979, 488)
• The function of proper names is simply to refer to objects no matter
how they are described - their function is to pick out the same object
(under our current usage of words) in all possible worlds where that
object exists
• Similar considerations apply to theoretical terms, namely they also
refer rigidly
3. Contingent identity
• Allan Gibbard (1975) – counterexample to 4)
• Lumpl (piece of clay) and Goliath (statue)
• „I make a clay statue of the infant Goliath in two pieces, one the part above
the waist and the other the part below the waist. Once I finish the two halves,
I stick them together thereby bringing in to existence simultaneously a new
piece of clay and a new statue. A day later I smash the statue, thereby
bringing to an end both statue and piece of clay. The statue and the piece of
clay persisted during exactly the same period of time.” (Gibbard 1975, 191)
• Lumpl=Goliath
• „suppose I had brought Lumpl in to existence as Goliath,just as I actually did,
but before the clay had a chance to dry, I squeezed it into a ball. At that point,
according to the persistence criteria I have given, the statue Goliath would
have ceased to exist, but the piece of clay Lumpl would still exist in a new
shape. Hence Lumpl would not be Goliath, even though both existed”
(Gibbard 1975, 191)
• Lumpl = Goliath & ◊(Lumpl exists & Goliath exists & Lumpl≠Goliath).
Contingent identity and proper names
• sortal theory of proper names
• it does not make a sense to ask “what that thing would be, (…), a part from
the way it is designated”, because “(p)roper names like 'Goliath or 'Lumpl
refer to a thing as a thing of a certain kind:' Goliath' refers to something as a
statue; 'Lumpl as a lump.” (Gibbard 1975, 194-195)
• Goliath refers to a thing qua statue
• Lumpl refers qua lump of clay
• Questions of identity through possible worlds only make sense when
we relativize them to a sortal
Rigid designators in the sortal theory
• „A designator may be rigid with respect to a sortal: it maybe statue-rigid, as
'Goliath' is, or it maybe lump-rigid as 'Lumpl' is. A designator for instance is
statue-rigid if it designates the same statue in every possible world in which
that statue exists and designates nothing in any other possible world. What is
special about proper names like 'Goliath' and ‘Lumpl’ is not that they are rigid
designators it is rather that each is rigid with respect to the sortal it invokes.
‘Goliath' refers to its bearer as a statue and is statue rigid; 'Lumpl' refers to its
bearer as a lump and is lump-rigid.” (Gibbard 1975, 195)
Persistence criteria
• What explains the possibility of contingent identity is the fact that the
sortals through which things are picked out invoke persistence criteria
• Lumpl qua lump of clay and Goliath qua statue have different
persistence criteria
• Reference of a term is determined by the origin of the object qua
something and by the persistence criteria
Persistence criteria
• Persistence criteria for pieces of clay:
“(a) The domain of P is an interval of time T.
(b) For any instant t in T, P(t) is a portion of clay the parts of which, at t,
are both stuck to each other and not stuck to any clay particles which
are not part o f P(t).
(c) The portions of clay P(t) change with t only slowly, if at all.
(d) No function P * which satisfies (a), (b), and (c) extends P, in the
sense that the domain of P* properly includes the domain of P and the
function P is P* with its domain restricted.” (Gibbard 1975, 189)
• a statue (e.g. Goliath) is identical to a lump of clay with a certain
shape
3.1. Contingent Identity and Leibniz’s law
• The problem with the sortal account of proper names is that it
seemingly violates Leibniz’s law of identity
15. □(Lumpl exists → Lumpl=Lumpl) – necessary truth
16. Goliath=Lumpl  contingent identity
17. □ (Lumpl exists → Goliath=Lumpl)  false
• However, by Leibniz’s Law 17. follows from 15. and 16.
Gibbard’s solution
• Leibniz’s law applies to properties and relations (ibid., 201).
Therefore, if we are going to apply the law in the present context one
needs to argue that modal contexts attribute properties to things
• Gibbard follows Quine in claiming that concrete things do not have
modal properties:
• „Modal expressions do not apply to concrete things independently of the way
they are designated. Lumpl, for instance, is the same thing as Goliath: it is a
clay statue of the infant Goliath which I put together and then broke.
Necessary identity to Lumpl, though, is not a property which that thing has or
lacks, for it makes no sense to ask whether that thing, as such, is necessarily
identical with Lumpl. Modal contexts, then, do not attribute properties or
relations to concrete things-so the proponent of contingent identity can
respond to Leibniz's Law.” (Gibbard 1975, 201-202)
Individual concepts
• Frege and Carnap
• In modal contexts names and variables shift their reference to their normal sense
18. □(Lumpl exists → Goliath=Lumpl)
• In 18) ‘Goliath’ and ‘Lumpl’ to the concept of Goliath and Lumpl
19. (Lumpl exists and x≠Lumpl)
• x does not range over concrete objects; rather it ranges over individual concepts.
• Individual concepts are functions from possible worlds whose values are objects
in those possible worlds.
References
• Burgess, John P. 2014. "On a derivation of the necessity of identity." Synthese 191: 1567–1585.
• Hayaki, Reina. 2006. "Contingent objects and the Barcan formula." Erkenntnis 64: 75-83.
• Burnyeat, Myles. 1979. "Saul Kripke: Identity and Individuation." In Philosophy as it is, edited by Ted
Honderich and Myles Burnyeat, 467-477. London: Penguin books.
• Gibbard, Allan. 1975. "Contingent Identity." Journal of Philosophical Logic 4: 187-221.
• Kripke, Saul. 1979. "Identity and necessity." In Philosophy as it is, edited by Ted Honderich and Myles
Burnyeat, 476-513. London: Penguin Books ltd.
• Kripke, Saul. 1971. "Identity and Necessity." In Identity and Individuation, edited by Milton K. Munitz, 135-164.
New York: New York University Press.
• Marcus, Ruth Barcan. 1946. “A functional calculus of first order based on strict implication.” Journal of
Symbolic Logic 11: 1-16.
• Quine, Willard V. O. 1961. "Reference and Modality." In From a Logical Point of View, by Willard V. O. Quine,
139-59. New York: Harper and Row.
• Wiggins, David. 1965. "Identity statements." In Analytic philosophy: Second series, edited by R. J. Butler, 40–
71. Oxford: Basil Blackwell.
• Williamson, Timothy. 2002. “Necessary Existents.” In Logic, Thought, and Language, edited by Anthony
O’Hear, 233-251. Cambridge: Cambridge University Press.